Robustness under Bounded Uncertainty with Phase Information A. L. Tits Dept. of Elec. Eng. and Inst. for Syst. Res. University of Maryland College Par

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1 Robustness under Bounded Uncertainty with Phase Information A L Tits Dept of Elec Eng and Inst for Syst Res University of Maryland College Park, MD 74 USA V Balakrishnan School of Elec and Comp Eng Purdue University West Lafayette, IN USA L Lee Dept of Elec Eng National Sun Yat-Sen Univ Kaohsiung, Taiwan 844 ROC February 1, 1998 Abstract We consider uncertain linear systems where the uncertainties, in addition to being bounded, also satisfy constraints on their phase In this context, we dene the \phase-sensitive structured singular value" (PS-SSV) of a matrix, and show that sucient (and sometimes necessary) conditions for stability of such uncertain linear systems can be rewritten as conditions involving PS-SSV We then derive upper bounds for PS-SSV, computable via convex optimization We extend these results to the case where the uncertainties are structured (diagonal or block-diagonal, for instance) 1 Introduction A popular paradigm for modeling control systems with uncertainties is illustrated in Fig 1 Here P (s) is the transfer function of a stable linear system, and is a stable operator that represents the \uncertainties" that arise from various sources such as modeling errors, neglected or unmodeled dynamics or parameters, etc Such control system models have found wide acceptance in robust control; see for example [1,, 3, 4] From the physical laws governing the system and from the modeling procedures used to arrive at the paradigm in Fig 1, the uncertainty is usually known or assumed to possess various additional properties Common examples are that is structured (ie, diagonal or block-diagonal), that it is linear time-invariant or real-constant etc Often, information about the size of (usually as a bound on some induced norm) is available For example, if is LTI, frequency response measurements can be used to estimate bounds on the L gain of It is also natural in some situations to assume that is dissipative or passive, ie, that it always dissipates energy Such can be the case, for example, with high order mechanical systems when the dynamics associated with a (poorly known) passive subsystem (eg, containing no energy sources and whose input and output are power-conjugate) are Research supported in part by NSF's Engineering Research Center Program, under grant NSFD-CDR The work of the third author was also supported in part by the National Science Council of Taiwan, ROC, under grant NSC E

2 - P (s) 1 6 Figure 1: Closed-loop system left out of the nominal model For example, it is mentioned in [5, 6] that a good model for lightly damped exible structures with colocated force actuators and rate sensors can be obtained in the paradigm in Fig 1, where P (s) is the transfer function of the nominal system model based on a few lower modal frequencies and mode shapes, with the higher modal frequencies and the corresponding mode shapes lumped together as a passive LTI uncertainty (s) (Also see Example 1 in x5) Perhaps the most fundamental question that arises with the feedback system in Fig 1 is that of \robust stability": Is the system stable for all possible instances of This question can be partially answered using a number of approaches For example, if the L -gain of does not exceed one, the small gain theorem [7] asserts that the system is robustly stable if the L gain (which is also the H 1 norm) of P is less than one And, if is known to be passive, the passivity theorem [7] asserts that closed-loop stability is ensured as long as P is strictly passive If it is known further that is block diagonal, then it is enough that an appropriately scaled version of P have L -gain less than one or be strictly passive, respectively Necessary and sucient conditions in this context can be expressed in terms of the structured singular value (see for example [8]) Suppose now that is known to be passive and, at the same time, to have an L -gain no larger than one Of course, either the small gain theorem or the passivity theorem can guarantee robust stability in this case, but intuitively, either approach alone would be too conservative, since in either case a seemingly important attribute of the uncertainty model is being ignored One objective of this paper is to address this issue It is often the case that the uncertainty is linear, time-invariant (LTI), and diagonal In this case, the unity-bounded L -gain and passivity assumptions on can be interpreted as knowledge on the frequency response of each diagonal entry ii of : the Nyquist plot of ii lies inside the unitdisk and in the right-half complex plane, respectively There are instances where it is appropriate to model ii as having its Nyquist plot entirely contained within some acute sector, of aperture < Such a sector can be assumed, without loss of generality (via simple loop transformation), to be a proper subset of the right-half plane This can occur when modeling is done from experimental data and the \Nyquist cloud" is better approximated by a sector portion of a disk than by a full disk; see Example 3 in x5 It can also occur when the uncertainty, due to several uncertain parameters, is \lumped" into a single dynamic uncertainty block; in this case, the approach presented in this paper would result in signicant computational savings in comparison with the direct approach; see

3 Example in x5 In both instances conservativeness can often be further reduced by allowing for frequency dependent sectors Investigation of robust stability under such \uncertainty with phase information" is a further objective of this paper Uncertainty is often best represented by a set of full matrices (or block-diagonal matrices), and handling this situation in our framework necessitates a concept of \phase of a matrix" Several authors have proposed such concepts In [9], the \principal phases" of a matrix are dened as the arguments of the eigenvalues of the unitary part of its polar decomposition, and a \small phase theorem" is derived that holds under rather stringent conditions Hung and MacFarlane, in [1], propose a \quasi-nyquist decomposition" in which the phase information of a transfer matrix is obtained by minimizing a measure of misalignment between the input and output singular vectors Finally, Owens, in [11], uses the numerical range to characterize phase uncertainty in multivariable systems The concept of phase we adopt here is related to that of [11] Our denition not only serves to characterize phase uncertainty in multivariable systems, but also provides a practical and tractable way of using uncertainty phase information in robustness analysis Thus, in this paper, we consider the robust stability of the system in Fig 1 when is a blockdiagonal LTI uncertainty that simultaneously satises constraints on its norm, and on its \phase" In x, we dene the phase-sensitive structured singular value (PS-SSV), dening in the process the phase of a matrix We then derive a condition for robust stability of the system in Fig 1 in terms of the PS-SSV It turns out that when the uncertainty is scalar, or made of diagonal scalar blocks, the PS-SSV-based condition on robust stability is both necessary and sucient Computing the PS-SSV exactly turns out to be an NP-hard problem We therefore concentrate on computing an upper bound on the PS-SSV, in x3 In x4, we show that computation of this upper bound can be reformulated as a quasi-convex optimization problem; we discuss some schemes for its solution In x5, we demonstrate our results via three numerical examples, and we conclude with x6 Many of the ideas developed in this paper were adapted from earlier work by two of the coauthors and M K H Fan, see [1, 13, 14, 15] Results closely related to those of x31 were obtained independently by Eszter and Hollot [16] for the case when the phase bounds amount to a passivity constraint on the uncertainty Notation R, R+ and Re denote the sets of real numbers, nonnegative real numbers, and R [ f1g (one point-compactication of R), respectively C, C + and C +e denote the set of complex numbers, complex numbers with positive real part (ie, the open right-half complex plane), and C + [ f1g respectively H 1 denotes the set of scalar- or matrix-valued functions that are analytic and bounded in the open right half plane, and RH 1 denotes the set of functions in H 1 that are real rational H denotes the complex-conjugate transpose of H C mn For H C nn satisfying H = H, the notation H (H > ) means that H is positive-semidenite (positive-denite) 3

4 The phase-sensitive structured singular value 1 Scalar Case Let us rst consider the case when both the LTI system and the LTI uncertainty in Fig 1 have a single input and a single output Thus P (s) and (s) are scalar transfer functions, and to emphasize this, we rename them as p(s) and (s), respectively Let : C! (; ] be the usual phase of a complex scalar, with () dened to be Note that j()j is lower semicontinuous (and continuous outside every neighborhood of the origin) Given a complex scalar m and a real scalar [; ], let (m) be dened by (m) = (inf fjj : j()j ; 1 + m = g) 1 ; if the set over which the inmum is taken is nonempty, and (m) = otherwise (ie, (m) = jmj if j(m)j, and otherwise) Note that (m) is upper semicontinuous in (; m) Theorem 1 below shows that various properties of the closed-loop system depicted in Fig 1 can be assessed from the knowledge of (p(s)) on the imaginary axis, under various assumptions on Theorem 1 Let p H 1 be continuous over C +e, let : Re! [; ], and let = f H 1 : is continuous on C +e; kk 1 1; j((j!))j (!) 8! Reg: Suppose that (a) sup!re (!) (p(j!)) < 1 Then (1 + p) 1 H 1 for all and, if is upper semicontinuous, sup k(1 + p) 1 k 1 < 1: (1) Moreover, if is constant, then statements (a), (b) and (c) are equivalent, where (b) and (c) are as follows: (b) (1 + p) 1 H 1 for all and sup k(1 + p) 1 k 1 < 1 (c) (1 + p) 1 H 1 for all RH 1 \ and sup RH 1\ k(1 + p) 1 k 1 < 1 Proof: We rst prove by contradiction that (a) implies that (1 + p) 1 H 1 for all Thus suppose that, for some, (1 + p) 1 6 H 1 It follows from Cauchy's Principle of the Argument, using a simple homotopy (see, eg, Lemma 1 in [17, 18] for details) that there exists (; 1] and ^! Re such that 1 + (j^!)p(j^!) = : Since, it is clear that j(j^!)j 1 and j((j^!))j (^!) Thus (^!) (p(j^!)) 1, a contradiction To complete the proof of the rst claim, suppose that is upper semicontinuous and, proceeding again by contradiction, suppose that (1 + p) 1 H 1 for all but that, given any > there exist and! R such that j1 + (j! )p(j! )j < : 4

5 Let = (j! ) and note that, since, j( )j (! ): Since j j 1 it follows from compactness of the complex unit disk, continuity of p on jre, lower semicontinuity of jj and upper semicontinuity of that there exits ^ C +e and ^! Re such that j^j 1, j(^)j (^!) and 1 + ^p(j^!) = Thus (^!) (p(j^!)) 1, a contradiction To complete the proof of the theorem, rst note that the implication (b))(c) holds trivially Suppose now that is constant The implication (a))(b) has just been proven It thus remains remains to show that (c))(a) We again use contradiction Thus assume that sup!r e (p(j!)) 1: We show that, given any >, there exists ^ \RH 1 and ^! Re such that j1+^(j^!)p(j^!)j <, a contradiction Let ~! Re be such that (p(j~!)) 1 (since p is continuous on jre and () is upper semicontinuous, such ~! always exists) Thus, for some ~ C +, with j~j 1 and j(~)j, 1 + ~p(j~!) = Note that, if =, the claim holds trivially (take ^(s) = ~ for all s); thus assume that > Since p is continuous on jre, there exist ^! R n fg and ^ f C + : jj < 1; j()j < g such that j1 + ^p(j^!)j < It is shown in Appendix A that, under these conditions, there exists ^ \ RH 1 such that ^(j^!) = ^ This completes the contradiction argument Remark: The upper semicontinuity assumption on is indeed needed in order for (1) (uniform robust stability) to follow, as shown by the following example Let p(s) = (s + 1)=(s + ) Let ^! be the frequency at which the phase of p(j!) is largest (in the rst quadrant) and let ^ be its value Dene by (^!) = = and (!) = ^ for all other! It is readily checked that (!) (p(j!)) = for all! but that 1 + p(j!)(j!) can be made arbitrarily close to in the neighborhood of ^! Remark: The suciency part of Theorem 1 can be extended to handle more general uncertainty sets See remark immediately following Theorem 3 We leave open the question of necessity of condition (a) of Theorem 1 under relaxed assumptions on It may be necessary, eg, to require that (!) not approach zero too fast The matrix case with structure 1 Phase and phase-sensitive structured singular value As a rst step toward extending the results of x1 to matrix-valued P and, we propose a concept of phase of a matrix Given a complex matrix, let N () be its numerical range, ie, N () = fx x : C = fx C n : kxk = 1g and k k is the Euclidean norm This set is known to be convex The following denition is a slight modication of that used in [15] 5

6 Denition 1 Let 6= be a complex square matrix such that = intn () The median phase MP() of is the angle, with a range of (; ], between the positive real axis and the ray bisecting the smallest sector containing N () The phase spread PS() of is half the angle of this sector (see Fig ) We dene MP() and PS() to be PS MP Figure : Numerical range, median phase and phase spread Thus MP() (; ] and PS() [; =] Below we will refer to the pair (MP(); PS()) as phase information of If intn (), there is no phase information for Note that in the case of a complex number a = e j with > and (; ], the phase information of a is (; ) Also, the phase information of a matrix is invariant under multiplication of the matrix by a positive number, and if is Hermitian positive semidenite, its phase information is (; ) Finally, the phase information of a matrix is invariant under unitary similarity transformations (since the numerical range is) Median phase and phase spread are related to the concept of principal phases introduced by Postlethwaite et al in [9] Namely, for any square complex matrix, MP() PS() min () max () MP() + PS() where min () and max () are the minimum and maximum principal phases of, respectively This result, stated dierently, was obtained by Owens [11] (who also used the term \phase spread") For any matrix with = intn (), N e jmp() C + In other words, we can rotate the numerical range of any matrix for which = intn () so that it is contained in the righthalf complex plane With this in mind, we restrict our attention in the sequel to matrices with + (or equivalently N () C +) For such matrices, we next give alternate characterizations of the phase information; these will serve us well in our derivation of stability tests in the sequel 6

7 Given with +, of particular interest is the smallest sector (ie, one that subtends the smallest angle at the origin) in the right-half plane, symmetric about the real axis, that contains N () (The interest stems from the fact that in the sequel, we will consider uncertainties whose numerical range is known to lie in such symmetric sectors at every frequency) Let () be the angle subtended by this sector at the origin Evidently (see Fig ), () = max fmp() + PS(); (MP() PS())g : () We then have the following alternate characterization for () Lemma 1 Let C nn, with + Then, () = cot sup 1 b : + j ( ) 8 fb; bg : (3) Proof: From Fig it is clear that, for b > cot(()) (ie, b 6= and b 1 < tan(())), there exists ^v C n such that Re(^v ^v) < bjim(^v ^v)j ie, for some fb; bg, + j ( ) 6 : Moreover, with b = cot(()) (1; 1), it is clear from the gure that, for all v C n, Re(v v) jim(v v)j 8 fb; bg ie, + j ( ) 8 fb; bg: Finally, if () = then N () is a subset of the negative imaginary axis, ie, =, and thus the matrix inequality in (3) holds for every nite Remark: It is easy to verify that for any satisfying + with () >, we have + j ( ) for all [ cot (); cot ()]: Lemma is lower semicontinuous over f : + g Proof: Let [; =] and let f k g! ^, with k + k for all k, and lim sup ( k ) We show that (^), proving the claim If = = the result is obvious Thus suppose [; =) and let (; = ] For k large enough, ( k ) +, and thus, taking the cotangent of both side of (3) ( + > ), for k large enough, k + k j ( k k) 8 [ cot( + ); cot( + )]: 7

8 It follows that ^ + ^ j (^ ^ ) 8 [ cot( + ); cot( + )]; ie, (again using (3)), that cot((^)) cot( + ), ie, (^) + Since this holds for arbitrarily small >, the claim follows With an eye towards issues of robust stability with respect to possibly block-structured uncertainty, we now extend the denition of to handle block-diagonal structures Given positive integers k i, i = 1; : : : ; `, such that P k i = n, we dene the set of block-diagonal matrices with block sizes k i as = fdiag( 1 ; : : : ; `) : i C k i k i g: (4) We next dene as the following phase-constrained subset of : = fdiag( 1 ; : : : ; `) : i C k i k i ; ( i ) i g; (5) where = ( 1 ; : : : ; `) with, for i = 1; : : : ; `, i [; =] Note that + for all Denition The phase-sensitive structured singular value of M C nn given by (M) = (inff() : ; det(i + M) = g) 1 with respect to is if det(i + M) = for some, and (M) = otherwise Properties of Unlike the \standard" mixed, is clearly not invariant under change of sign of its argument Thus, in particular, it is not always larger than the spectral radius (complex ) or the real spectral radius R On the other hand it is clear that where, for any complex matrix M, R (M) (M) (M); (6) R (M) = maxf : is a negative, real eigenvalue of Mg; with R (M) = if M has no negative, real eigenvalues This leads to the following easily derived characterization of (Note that R is upper semicontinuous, which justies the \max") Theorem (M) = max ; ()1 R (M) = max ; ()1 R (M): (7) Like the standard mixed, is invariant under similarity scaling of its argument by matrices that commute with the elements of the uncertainty set, ie, given any nonsingular matrix D = diag(d 1 I k1 ; : : : ; d`i k`), (M) = (DMD 1 ): 8

9 In general however, is clearly not invariant under pre- or post-multiplication of its argument by a unitary matrix in Next, it is readily veried that (M) is monotonic nondecreasing in each of the components of and, using lower semicontinuity of (Lemma ), that (M) is upper semicontinuous in (; M) Finally, the following result holds Proposition 1 Let P H 1 be continuous over C +e, and let [; =]` Then sup (P (j!)) = sup (P (s)):!r e sc +e Proof: We show that the following statements are equivalent: (a) sup!r e (P (j!)) < 1; (b) (c) (I + P ) 1 H 1 8 ; () 1; sup sc +e (P (s)) < 1: Since is positive homogeneous, the claim then follows from the equivalence of (a) and (c) We rst show by contradiction that (a))(b) Thus let ^, with (^) 1, be such that (I + ^P ) 1 6 H 1 As in the proof of Theorem 1, it follows from Cauchy's Principle of the Argument that there exist (; 1] and ^! Re such that det(i + ^P (j^!)) = : Since ^ and (^) 1, this implies that (P (j^!)) 1, a contradiction Concerning the implication (b))(c), if there exists ^s C +e be such that (P (^s)) 1, then there exists ^, with (^) 1, such that det(i + ^P (^s)) =, contradicting (b) Finally, the implication (c))(a) holds trivially 3 The small- theorem Given any : Re! [; =]`, we dene = f H 1 : is continuous on C +e; kk 1 1; (j!) (!) 8! Reg: Theorem 3 Let P H 1 be continuous over C +e, let : Re! [; =]` Suppose that (a) sup!re (!) (P (j!)) < 1 Then (I + P ) 1 H 1 for all, and if is upper semicontinuous, then sup k(i + P ) 1 k 1 < 1: Moreover, if is constant, then (a) is equivalent to (b) (I + P ) 1 H 1 for all and sup k(i + P ) 1 k 1 < 1: 9

10 Proof: The implication (a))(b) is proved as in Theorem 1 with det(i + P ) replacing 1 + p Concerning the implication (b))(a), note that, if is constant and (a) does not hold, then (since P is continuous over C +e and is upper semicontinuous) there exists, among others, a constant (complex) and some ^! Re such that det(i + P (j^!)) =, contradicting (b) Remark: Again, the suciency part of Theorem 3 can be extended to handle more general uncertainty sets For example, consider the uncertainty set ~ = f H 1 : is continuous on C +e ; U(!)(j!) (!) ; ( i (j!)) d i (!); i = 1; : : : ; `; 8! R e g; where d i : R e! [; 1), i = 1; : : : ; `, : R e! [; =]`, and U(!) = diag(u 1 (!)I k1 ; : : : ; u`(!)i k`), with u i : R e! fz C : jzj = 1g Then, it is easy to show that if sup!r e (!) (diag(d i (!)I ki )U(!) P (j!)) < 1; then, (1 + P ) 1 H 1 for all ~ ; and that if, in addition, d i and are upper semicontinuous, and U is continuous, then sup k(1 + P ) 1 k < 1: ~ Again we will leave open the question of necessity of condition (a) of Theorem 3 under relaxed assumptions on On the other hand, even for constant, it is unclear in general whether, if (a) does not hold, there exists real on the real axis (which must be the case if is the transfer function of a real impulse response) such that (I + P ) 1 6 H 1 or k(i + P ) 1 k 1 is arbitrarily large In the case of purely diagonal uncertainty structures, though, this is the case even with the additional requirement that be rational In other words the following holds Theorem 4 Let P H 1 be continuous over C +e, let [; =]`, and suppose k i = 1, i = 1; : : : ; ` (= n), ie, suppose that consists of diagonal matrices The following statements are equivalent (a) sup!re (P (j!)) < 1 (b) (I + P ) 1 H 1 for all and sup k(i + P ) 1 k 1 < 1 (c) (I + P ) 1 H 1 for all RH 1 \ and sup RH 1\ k(i + P ) 1 k 1 < 1 (The proof of the implication (c))(a) is exactly along the lines of that of the corresponding implication in Theorem 1) 3 Upper Bounds on So far, we have seen denitions of, and how conditions on give sucient (and sometimes necessary) conditions for uniform robust stability In this section, we will concern ourselves with the numerical computation of Computing exactly is equivalent to nding the global minimum of a nonconvex optimization problem, and we are not aware of any ecient solution methods for it Therefore, we will not attempt to compute directly; instead, we will derive numerically computable upper bounds on, which will give, in turn, sucient conditions for robust stability 1

11 31 The matrix case with structure Computing (m) for a scalar m is trivial We then consider the problem of computing an upper bound on (M), when M is a matrix, and is assumed to have some structure, that is, it is required to belong to the set Let fs : S = fdiag(s 1 I k1 ; : : : ; s`i k`) : s i > g ; and, given = ( 1 ; : : : ; `) with i [; =], i = 1; : : : ; `, let ( B B = diag( = B : 1 I k1 ; : : : ; `I k`), i R, i = 1; : : : ; `, with i [ cot i ; cot i ] when i > ) : We then have the following lemmas Lemma 3 Let, with I Then, satises, for every R, S S, and B B, " I # " R (I jb)s S(I + jb) R # " I # : (8) Proof: Consider any satisfying I Since commutes with every R R, we have R R : (9) Next, since, in view of Lemma 1 and of the remark following it, and since B B whenever B B, we have for every B B, + j(b B) : Since, every S R and every B B commute with each other, we then have S + S j(bs SB) : (1) From (9) and (1), we conclude that every with I satises, for every R, S S and B B, R R + (I jb)s + S(I + jb) ; which is equivalent to (8) Theorem 5 Let, with I If there exists some R S, S S, and B B, such that then det(i + M) 6= M RM R (S(I + jb)m + M (I jb)s) < ; (11) Remark: Theorem 5 constitutes a special case of the general stability theorem for systems with uncertainties described by integral quadratic constraints or IQCs [19, Theorem 1] In particular, Theorem 5 can be viewed as a sucient condition for the well-posedness of a feedback interconnection of a constant matrix 11

12 with a constant phase- and norm-bounded uncertainty in the feedback loop Since there are no dynamics involved, a direct linear algebraic proof can be given, which we present next for the sake of completeness Proof: Rewriting (11) as " M I # " R (I jb)s S(I + jb) R # " M I # < ; (1) we now proceed by contradiction Suppose that det(i + M) = Then for some nonzero v C n, we have (I + M)v = Dening u = Mv, we have v = u Now, from (1), we have v " M I ie, " u v # " # " But from Lemma 3, we must have " # " I R (I jb)s S(I + jb) R R (I jb)s S(I + jb) R R (I jb)s S(I + jb)s R # " M I # " u v # " I # # v < ; < : # ; which yields (u) " I # " R (I jb)s S(I + jb) R # " I # (u) ; ie, " u v # " R (I jb)s S(I + jb) R # " u v # ; which is a contradiction We can use Theorem 5 to derive an upper bound ^ (M) on (M) Suppose that for some >, R and S in S, and some B in B, we have M RM R (S(I + jb)m + M (I jb)s) < : Then, it can be shown with a little algebra that is an upper bound on (M) We therefore have the following upper bound on (M) 1 Corollary 1 Let M C nn Then (M) ^ (M), where ^ (M) = inf ( : M RM R S(I + jb)m M (I jb)s < > ; R; S S; B B ) : (13) 1 This result was rst reported, in a slightly dierent form, in [1] For the case when i = =, i = 1; : : : ; ` (passive uncertainty), it is a special case of a result obtained independently by Eszter and Hollot [16] 1

13 The conclusion of Corollary 1 represents one of the central contributions of the paper we now have an upper bound for, which, as we shall see in x4, can be numerically computed quite eciently, using convex optimization techniques Remark: It is easily shown that, for any scalar m, ^ (m) = (m) 3 An o-axis circle-criterion interpretation As was done in [] and in xv of [1] in the context of the \classical" mixed, it is possible to obtain the phase-sensitive upper bound by optimizing the complex upper bound over a set of disk uncertainties Consider a \block-diagonal disk uncertainty set", ie, a set of block diagonal matrices such that each block ranges over a certain \hyperdisk", namely over the image of f i : ( i ) 1g under a certain linear fractional transformation A \complex-" type upper bound is readily obtained corresponding to such uncertainty blocks Clearly, if the uncertainty set covers f : () 1g, then this upper bound is also an upper bound for Below we show that minimizing this bound over a certain family of such transformations yields precisely the bound given by Theorem 5 and Corollary 1 Given S S and B B, let " # T = F (I + F F ) 1= I (I + F F ) 1= ; where F = S(I + jb) It is readily checked that the \lower" linear fractional transformation F l (T; M) is well dened for any M, that the \upper" linear fractional transformation F u (T; ) is well dened whenever () 1, and that (provided () 1) F l (T; M) = (F M)(I + F F ) 1= ; F u (T; ) = ((I + F F ) 1= F ) 1 : Consequently, the systems in the three block diagrams of Fig 3 are all equivalent in the sense that each one is well-posed if and only if the other two are 1-6 T M 1 - F u (T; ) 6 M - F l (T; M) Figure 3: Three \equivalent" block diagrams 13

14 For the sake of geometric intuition consider now the case of a diagonal (rather than block diagonal) structure, say, F = diag(f i ), with f i = s i (1 + j i ), s i > and j i j cot i when i > Let B be the set of complex diagonal matrices with () 1 The image of B under the linear fractional transformation F u (T; ) is given by ( F u (T; B) = diag 1q! ) jf i j f i : jj 1 : It is straightforward to check that each diagonal entry ranges over a circle of radius p 1 + jf i j = q 1 + s i (1 + i ) centered at f i = s i (1 j i ) (see Fig 4) It follows that, for each s i > and each i with j i j cot i when i >, f : () 1g F u (T; B); (14) which shows that each diagonal \disk" entry of F u (T; B) \covers" the corresponding entry in the uncertainty set of interest, f : () 1g Conversely, it is easy to show that any disk that covers a diagonal entry in the uncertainty set f : () 1g must be the corresponding diagonal entry of F u (T; B) for some T : it is easy to solve \backwards" for s i and i, given the center and radius of the disk The same inclusion (14) holds in the general (block diagonal) case Indeed T 1 = " (I + F F ) 1= I F and simple algebra shows that, for any with () 1, F l (T 1 ; ) is well dened and # : and thus it is enough to show that F u (T; F l (T 1 ; )) = ; F l (T 1 ; f : () 1g) B: To see that the latter inclusion holds, assume without loss of generality that ` = 1 (full matrix uncertainty), ie, F = fi, with f = s(1 + j), s >, jj cot when >, and let with () 1 It remains to show that (F l (T 1 ; )) 1; or, equivalently, I (1 + jfj )(I + f ) 1 (I + f) 1 ; ie, via a congruence transformation, (I + f )(I + f) (1 + jfj ) ; ie (I ) + (f + f ) : 14

15 Since this clearly holds for any with () 1, (14) holds in the general case as claimed A sucient condition for ie, for (M) < 1, is thus that det(i + M) 6= 8 ; () 1; det(i + M) 6= 8 F u (T; B): Since the second and third block diagrams in Fig 3 are equivalent, the latter holds if and only if det(i F l (T; M)) 6= 8 B; ie, (F l (T; M)) < 1; and a sucient condition for this is that, for some R S, (RF l (T; M)R 1 ) < 1: (15) Since S and B commute, letting M R = RMR 1, we get RF l (T; M)R 1 = (F M R )(I + F F ) 1= : It follows that (15) holds if, and only if ((F M R )(I + F F ) 1= ) ((F M R )(I + F F ) 1= ) < I; ie, if, and only if, which holds if, and only if, (F M R)(F M R ) < I + F F; M R M R I F M R M R F < ; which is equivalent to the condition given in Theorem 5 33 Some special cases It is instructive to study the application of Theorem 5 and Corollary 1 to some special cases for the set These cases are encountered more often in practice; also, for some of these special cases, we can relate our results to those from literature 331 Bounded passive uncertainty We consider rst the case when the consists of unstructured or full matrices (ie, ` = 1) with a known bound on their maximum singular value, and whose phase is known to be = or less This situation arises when the uncertainty is passive and bounded If were scalar (ie, k 1 = 1), this 15

16 (1) (1) () () (3) (3) (4) (4) (5) (5) (1): i = cot( i ) (): cot( i ) < i < (3): i = (4): < i < cot( i ) (5): i = cot( i ) (1),(): i = cot( i ) (4),(5): i = cot( i ) (1),(5): s i = 1 (),(4): < s i < 1 (3): s i = Figure 4: Covering the uncertainty with o-axis circles The gure on the left shows the covering of the ith diagonal phase-bounded uncertainty with disk-uncertainties obtained by loop transforming the unit-disk with f i = (1 + j i ) for various values of i The gure on the right shows the covering with disk-uncertainties obtained by loop transforming the unit-disk with f i = s i (1 j cot i ) for various values of s i would mean that the Nyquist plot of is in a semicircle of known radius that lies in the right-half complex plane, shown in in Fig 5(a) In this case, B = fg and S consists of positive multiples of the identity matrix Therefore, from Corollary 1, we have which further simplies to ^ (M) = inf ( : rm M ri s (M + M ) < > ; r > ; s > ^ (M) = (max f; inf f max (M M c (M + M )) : c > gg) 1= ; where max denotes the largest eigenvalue of the corresponding Hermitian matrix 33 Bounded, constant, Hermitian, positive-denite uncertainty We next consider the case when is a constant, Hermitian, positive-denite matrix, with a known bound on its maximum singular value If the uncertainty were scalar (ie, k 1 = 1), this would mean that the Nyquist plot of the uncertainty is simply a point in a sub-interval of the positive real axis, as shown in Fig 5(b) ) ; 16

17 (a) Bounded passive uncertainty: The Nyquist plot is known to lie in the shaded region (b) Positive, real uncertainty: The Nyquist plot is known to lie in a subinterval of the positive real axis Figure 5: Various special cases In this case the set B consists of arbitrary real multiples of the identity, while S consists of positive multiples of the identity Therefore, we have, ^ (M) = inf which further simplies to ( : rm M ri s(1 + jb)m M (1 jb)s < > ; r > ; s > ; b R ^ (M) = (max f; inf f max (M M (c + jd)m M (c jd)) : c > ; d Rgg) 1= : It is instructive to consider other special cases of the instances considered above, when the uncertainty is diagonal, so that k 1 = = k` = Diagonal bounded passive uncertainty Suppose that the Nyquist plot of each of the diagonal uncertainties is known to lie a half-disk such as the one shown in Fig 5(a) In other words the uncertainty is diagonal, passive and bounded In this case, the set B = fg and the set S consists of diagonal positive-denite matrices Here ^ (M) = inf ( : M RM R SM M S < > ; R; S > and diagonal 334 Diagonal, bounded, positive, constant real uncertainty Finally, we consider the case when each of the diagonal uncertainties is a constant unknown parameter, known only to lie in some sub-interval of the positive real axis such as the one shown in Fig 5(b) Such uncertainties are often called parametric uncertainties Here, the set B consists of arbitrary real diagonal matrices, while S consists of diagonal positive-denite matrices Thus, ^ (M) = inf ( : M RM R S(I + jb)m M (I jb)s < > ; R; S; B real and diagonal, R >, S > 17 ) : ) ; ) : (16)

18 Remark: This case of bounded diagonal real uncertainty is well-studied in the literature, usually under the name of \real-" analysis; see for example, [1, ] The problem considered in these references is the computation of R(M), which is dened as < = () : is diagonal and real, A if det(i + M) = for some R(M) = : ; det(i + M) = diagonal and real, >: otherwise We point out that R(M) is dierent from (M) with = (for ease of reference, we will call the latter quantity R + and its upper bound given in (16) by ^R + ) The dierence between R and R + is that with R +, the uncertainty is required to be nonnegative, unlike with the denition of R For this reason, we will refer to R as \two-sided real-", while we will call R + \one-sided real-" The upper bound for the two-sided real- from [1] and [] can be easily adapted via a loop transformation to yield an upper bound for the one-sided real- This upper bound on R + is just ~R + (M) = inf : S S(I + jb)m M (I jb)s < > ; S; B diagonal, S > : (17) Remarkably, computing ~R + using (17) has the same complexity as computing ^R + using (16) Extensive numerical simulations suggest that this upper bound is tighter than the bound (16) We should note however that the bound (17) does not extend to the case of general phase-bounded uncertainty considered in this paper Finally, we note that it is possible to adapt ^R +, the upper bound for the one-sided real, to yield an upper bound for the two-sided real This upper bound on R turns out to be inf : M (3R + S) M R + ((S R + jb) M M (S R jb)) < > ; S; B diagonal, S > : However, we know of no ecient way of computing this upper bound 4 Computing sup! ^ (!) (P (j!)) From Theorem 3 in x3, it should be clear that the computation of sup!re (!) (P (j!)), which we shall denote by M (P ), is of considerable interest For reasons pointed out at the beginning of x3, we will consider instead the problem of computing sup! ^ (!) (P (j!)), which we shall denote by ^M (P ) Since ^M (P ) M (P ), computing ^M (P ) will enable us to state sucient conditions for the stability of the system in Fig 1 For each frequency!, the quantity ^ (!) (P (j!)), dened in Corollary 1, can be computed as the solution to a quasi-convex optimization problem There are several ways of showing this; we will demonstrate one method For convenience, we let M = P (j!) Recall that ^ is given by (13) Let T = BS Then the condition on B is equivalent to where is a constant diagonal matrix given by S > T > S; = diag (cot ( 1 (!)) I k1 ; ; cot (`(!)) I k`) : 18

19 Thus ^ is given as the optimal value of obtained by solving the problem minimize subject to R > M RM (SM + M S) j(t M M T ) S > T > S; R = diag(r 1 I k1 ; : : : ; r`i k`); r i > (18) S = diag(s 1 I k1 ; : : : ; s`i k`); s i > T = diag(t 1 I k1 ; : : : ; t`i k`) With =, the optimization variables in this problem are, R, S and T Problem (18) is one of minimizing a linear objective, subject to constraints on, R, S and T that are convex (in fact, linear matrix inequalities ) in R, S and T for xed, and vice versa It can be shown that problem (18) is a quasi-convex optimization problem [3] Much work has been done lately on problems such as (18): it is well-known that such problems have polynomial worst-case complexity; moreover, very ecient algorithms and software tools are available for their solution [4, 5] Next, we have the following obvious lower bound on ^M (P ) Lemma 4 Let = f! ;! 1 ; : : : ;! N g be a set of frequencies Then, ^M lb (P; ) = max ^ (P (j! (!i i i)) ; ) ^M lb (P; ), dened as satises ^M lb (P; ) ^M (P ), ie, it is a lower bound on ^M (P ) In order to compute ^M lb (P; ), we need to solve N + 1 quasi-convex optimization problems of the form (18) Of course, the number and choice of frequencies comprising determines how tight a bound ^M lb (P; ) is Remark: The lower bound given by Lemma 4 suers from a possible shortcoming: It is known that in general, ^ (!) (P (j!)) may be discontinuous as a function of! Specically, ^ (!) (P (j!)) might only be upper semicontinuous, and therefore we have no guarantees with the convergence of the lower bound ^M lb (P; ) to ^M (P ) even if N, the number of elements of, tends to 1 (but a scheme analogous to that proposed in [6] might be applicable) However, in most engineering applications (as we will see in x5), this does not pose a serious problem It is also possible to compute upper bounds on ^M (P ) using state-space methods The basic idea is this ^M (P ) if and only if there exist R : jr! R nn, S : jr! R nn and T : jr! R nn such that for A linear matrix inequality or an LMI is a matrix inequality of the form F (x) = F + P m where F i are given Hermitian matrices, and the x is are the real optimization variables i=1 xifi > or F (x), 19

20 every! R e, the following constraints are satised (the dependence of on! is now made explicit) (i) (ii) R(j!) > P (j!) R(j!)P (j!) (S(j!)P (j!) + P (j!) S(j!)) (!)S(j!) > T (j!); j(t (j!)p (j!) P (j!) T (j!)); (iii) T (j!) > (!)S(j!); (iv) R(j!) = diag(r 1 (j!)i k1 ; : : : ; r`(j!)i k`); r i (j!) > (19) (v) S(j!) = diag(s 1 (j!)i k1 ; : : : ; s`(j!)i k`); s i (j!) > (vi) T (j!) = diag(t 1 (j!)i k1 ; : : : ; t`(j!)i k`) It can be shown [7] that the constraints in (19) hold for some if and only if they hold for some realrational transfer functions ^R, ^S and ^T This fact can be combined with the Positive-Real (PR) lemma [8, 9] to write down LMIs whose feasibility is equivalent to conditions (i){(v) (see for example, [, 7] for an illustration of this procedure) Thus, a sucient condition for the feasibility of problem (19) can be recast as an LMI feasibility problem A bisection scheme can then be used to compute an upper bound for ^M (P ) It is also possible to avoid the bisection scheme altogether, by recasting the upper bound computation problem as a single generalized eigenvalue minimization problem; see [3] 5 Numerical examples We demonstrate on a few examples the application of stability tests based on the PS-SSV 51 Example 1: Stability of a exible structure We consider the stability of a planar truss structure, with a model adapted from the one presented in [5] The truss structure has sixteen free nodes, each with two degrees of freedom; thus it exhibits thirty-two exible modes We assume that the rst mode is exactly modeled as a linear time-invariant system, with transfer function p given by p(s) = :397 s s + (:75)(131)s : The remaining modes are modeled as a linear time-invariant uncertainty, with transfer function denoted by (s) It is known that is stable, and satises j(j!)j :337; Re (j!) ; for all! R; () that is, is passive, and has an H 1 norm bound of :337 A linear time-invariant controller c with transfer function c(s) = :38s5 + 33:18s :s :1s + 396:51s :16 s :15s :81s :36s :79s :7 has been designed to stabilize p(s), placing the poles at 1, 4 and 1 The robust stability question then is whether the controller stabilizes p +

21 jg(j!)j Re g(j!) - f c(s) - (s) p(s) - f g(s) (s) (a) Block diagram of the exible truss structure (b) Block diagram redrawn in our framework Figure 6: Example 1: Models of the exible truss structure ! (rad/sec) (a) Magnitude of g ! (rad/sec) (b) Real part of g Figure 7: Example 1: Frequency response of g The block diagram of the system is shown in Fig 6(a) The system redrawn in our analysis framework is shown in Fig 6(b), where g = c=(1+pc) The magnitude and real part of g are shown in Fig 7 From an inspection of these plots, and the properties of given in (), we conclude that: The small gain theorem does not prove stability of the system in Fig 6(b), since the H 1 norm of g exceeds 1=:337 The passivity theorem does not prove stability of the system in Fig 6(b), since g is not strictly passive (the real part of g(j!) is nonpositive for some!) However, the analysis techniques presented in this paper do prove uniform robust stability A plot of (g(j!)) is shown in Fig 8 (Since g is a scalar transfer function, is trivial to compute) Since sup!re (g(j!)) < 1=:337, the system in Fig 6(b) is indeed uniformly robustly stable 1

22 (g(j!)) ! (rad/sec) Figure 8: Example 1: The PS-SSV of g(j!) versus! 5 Example : Analysis of parametric systems We next consider the problem of uniform robust stability of the closed-loop system shown in Fig 9(a) P is the parameter-dependent plant, with transfer function given by P (s) = diag(p 1 (s); p (s)); p i (s) = a i(s + b i ) s + c i s + 1 ; a i [; 1]; b i [1; ]; c i [1; ]; and C is the controller with the transfer function s + s 1 (s C(s) = :3 + 1)(s + ) s + 1 s + 7 s : s + 1 The problem now is to ascertain the stability of this system for all allowable values of the parameters f C(s) P (s) C(s) P (s) (a) Block diagram of the parameter-dependent system (b) Block diagram redrawn in our framework Figure 9: Example : Stability analysis of a parameter-dependent system

23 Bound on jp i (j!)j (in degrees) Bound on j(p(j!)) j (deg) Fig 1 shows the values of the frequency response of p i, over a number of allowable parameter values, at a sample list of frequencies This gure indicates that each p i is passive, and has a frequency response which can be described as satisfying certain magnitude and phase constraints Fig 11 shows the magnitude and phase constraints on each of the terms! = :1 rad/sec! = :5 rad/sec! = :1 rad/sec! = : rad/sec! = :5 rad/sec! = 1 rad/sec! = rad/sec! = 5 rad/sec Figure 1: Example : Frequency response of each p i at a number of frequencies ! (rad/sec) (a) Magnitude constraints on p i(j!) ! (rad/sec) (b) Average value of (p i(j!)) ! (rad/sec) (c) Magnitude of variation of (p i(j!)) about its average Figure 11: Example : Magnitude and phase constraints on p i (j!) This problem can be posed in our PS-SSV framework, as shown in Fig 9(b) The uniform robust stability condition is sup (P (j!)) (!) e j (!) C(j!) < 1; (1)!R e 3

24 where (!) and the entries of (!) are plotted against! in Figs 11(b) and 11(c), respectively For convenience we let ~ C(j!) = e j (!) C(j!) A plot of ^ (!) ( ~ C(j!)) is shown in Fig 1, in solid lines For reference, the optimally scaled maximum singular value of ~ C(j!) is shown in dotted lines; this is an upper bound on ( ~ C(j!)), which can be thought of as an upper bound on PS-SSV that does not use the phase information Since condition (1) holds, the system in Fig 9(b) is indeed uniformly robustly stable Note that, since (P (j)) =, the bound on PS-SSV that does not use the phase information does not yield this conclusion ! (rad/sec) Figure 1: Example : The upper bound on (!) ~C(j!) is plotted against! in solid lines The optimally scaled maximum singular value of ~ C(j!) is plotted against! in dotted lines Remark: There is a more direct method of analyzing parameter-dependent systems, namely \real-" analysis (see [1]) It is of interest to compare PS-SSV-based stability methods with real- methods Let us consider the question of whether the system in Fig 9(b) is uniformly robustly stable The answer is armative in the PS-SSV framework if sup!r e (P (j!))^ (!) ~C(j!) is less than one Checking this numerically, from the discussion in x4 (in particular, Lemma 4), requires the solution of N LMI feasibility problems, one for each frequency Let us consider one such feasibility problem The variables in this problem are diagonal matrices R, S and T Thus, the number of scalar variables is 6 There is one LMI constraint of size, and 6 scalar constraints When the uniform robust robust stability of the same system is posed in the real- framework of [1], we once again have to solve an LMI feasibility problem at each frequency Here the variables in each problem are diagonal 6 6 matrices D = D T and G = G T (see [1] for details); thus the number of scalar optimization variables is 1 There is one LMI constraint of size 6 6, and 6 scalar constraints For the problem of uniform robust stability with parametric uncertainties, PS-SSV-based tests are likely to be more conservative than real- tests However, it should be clear from the number of variables and constraints that the amount of computation required by PS-SSV-based methods is less than that required by real- methods For our example, empirical studies indicate that the computation required by real- 4

25 degrees max () methods is approximately 1 times that required by PS-SSV-based methods [31] 3 Thus, the PS-SSV approach can be useful in analyzing parameter-dependent systems, albeit more conservatively, when the number of parameters is large 53 Example 3: Experimentally measured matrix phase information We consider an uncertain system as in Fig 1, where the plant P is strictly proper (ie, P (1) = ), has two inputs, two outputs, and a state-space realization (A; B; C) with A = ; B = ; C = 1 " # 1 1 : We assume that the two-input two-output LTI uncertainty has been experimentally measured at a number of frequencies A scatter-plot of the phase information of (j!) at a number of frequencies is shown in Fig 13(a); a scatter-plot of the norm of (j!) at a number of frequencies is shown in Fig 13(b) ! (rad/sec) (a) Scatter plot of the MP((j!)) + PS((j!)) (denoted by \+") and MP((j!)) PS((j!)) (denoted by \"), for various, versus! ! (rad/sec) (b) Scatter plot of the norm of (j!) (for various ) versus! Figure 13: Example 3: Experimentally determined magnitude and phase characteristics of From the scatter plot shown in Fig 13(a), we can determine continuous functions lb and ub such that for every frequency! and, the smallest sector containing N ((j!)) is n o z : z = re j ; r ; [ lb (!); ub (!)] : (These functions are shown in solid lines in Fig 13(a)) Also, from Fig 13(b), we can determine a function d(!) such that for every frequency! and, ((j!)) < d(!): 3 In general, for an LMI problem with k variables and L LMI constraints of size n i ni, the computation required is dominated by O k P L i=1 ni(ni + 1)= 5

26 (This function is shown in a solid line in Fig 13(b)) Then, dening (!) = :5( ub (!) lb (!)) and (!) = :5( ub (!) + lb (!)), we have that the system in Fig 1 is uniformly robustly stable if sup (!) e j (!) P (j!) d(!) < 1;!R e where in the notation of x1, k 1 =, and = () The upper bound ^ (!) (P (j!)e j (!) ) from (13) is obtained for various! by solving the optimization problem (18), and plotted in Fig 14 Since sup!re ^ (!) (e j (!) P (j!))d(!) < 1, the system in Fig 1 is indeed uniformly robustly stable ! (rad/sec) Figure 14: Example 3: Upper bound on (!) (e j (!) P (j!)) as a function of! 6 Conclusions The \phase-sensitive structured singular value" framework developed in this paper provides an eective robustness analysis tool in various situations, eg, in the case when the uncertainty, besides being (possibly block-structured and) small, is known to be passive Several issues have been left unresolved 1 Under what \minimal" assumptions on () are statements (a), (b) and (c) of Theorem 1 equivalent In the presence of non-scalar \full blocks", and with constant, are statements (a) and (b) in Theorem 3 equivalent to the analogue of statement (c) in Theorem 1, namely (c) (I + P ) 1 H 1 for all RH 1 \ and sup RH 1\ k(i + P ) 1 k 1 < 1: 6

27 3 When is the upper bound ^ dened in Corollary 1 of x3 equal to, in particular is it always equal to when ` = 1 (full block uncertainty) The answer to some of these questions may be within reach The contributions in the paper can be generally viewed as the following: When the uncertainty in Fig 1 is LTI, and when additional information on the phase of the frequency response of is available, we have derived sucient (and sometimes necessary) conditions for robust stability A natural extension of this problem considered in this paper is the following Consider for simplicity the case when is a scalar uncertainty, and suppose that it is known that the Nyquist plot of is restricted to lie in some region in the complex plane that can be described as the intersection of generalized disks (ie, disks and half-spaces) Then, we can derive a sucient robust stability condition by combining robust stability conditions for each generalized disk, just as we did to arrive at Theorem 5 As a further extension along these lines, consider the situation when the Nyquist plot of is restricted to lie in some region in the complex plane that can be described as the union of sets which are themselves obtained as an intersection of generalized disks (A classic example of such a region is the \buttery" uncertainty set, described in [14]) The techniques described in this paper can be extended to handle these more general cases as well The focus of this paper has been exclusively on uncertainties about which phase information is available The techniques herein can be combined with other standard robustness analysis techniques such as complex or real- analysis, when phase information about only certain blocks of the uncertainty is available, leading to a new \mixed-" paradigm Finally, while the theory was developed for the continuous-time case, extension to discrete time is straightforward Appendix A Proposition Let (; ], let ^! R n fg, and let C + be such that jj < 1 and j()j < There exists RH 1, continuous on C +e, such that (j^!) = and such that kk 1 < 1 and sup!r j(((j!)))j < Proof: If =, simply let map C + to zero Assume now 6= Let D = fz C : jzj < 1g and let D = fz D : Re z ; j(z)j < g We rst construct a non-rational mapping ~ : D! C, taking real values on the real axis, such that (D) ~ belongs to D and contains and 1= + j in its interior This map is selected from a one-parameter family of mappings ~ : D! C, (; 1), constructed as the composition of two maps, ie, ~ = ~ ~ 1 First, for (; 1), the map ~ 1, dened on D, is given by 1 + z (1 z cos + (z) ~ 1 (z) = ) 1= ; with sin ( =) = 1 + z + (1 z cos + (z) ) 1= : For xed, ~ 1 maps D to the interior of a set such as the one depicted in Fig 15(a) Next ~, dened on ~ 1 (D), is given by ~ (w) = (w + (1 )) = : 7